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Liouville Type Results for Two-Sided Hypersurfaces in Weighted Killing Warped Products

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Abstract

We establish Liouville type results concerning two-sided hypersurfaces immersed in a weighted Killing warped product, under suitable constraints either on the Bakry-Émery-Ricci tensor of the base of the ambient space or on the height function of the hypersurface.

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References

  • Alías, L.J., Dajczer, M., Ripoll, J.: A Bernstein-type theorem for Riemannian manifolds with a Killing field. Ann. Glob. Anal. Geom. 31, 363–373 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  • Barbosa, J.L.M., do Carmo, M., Eschenburg, J.: Stability of hypersurfaces with constant mean curvature. Math. Z. 197, 123–138 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  • Bernstein, S.: Sur les surfaces d’efinies au moyen de leur courboure moyenne ou totale. Ann. Ec. Norm. Sup. 27, 233–256 (1910)

    Article  Google Scholar 

  • Bakry, D., Émery, M.: Diffusions hypercontractives. In: Seminaire de probabilites, XIX, 1983/84, volume 1123 of Lecture Notes in Math. Springer, Berlin, pp 177–206 (1985)

  • Bessa, G.P., Pigola, S., Setti, A.G.: On the \(L^1\)-Liouville property of stochastically incomplete manifolds. Potential Anal. 39, 313–324 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  • Charalambous, N., Lu, Z.: The \(L^1\) Liouville property on weighted manifolds. Contemp. Math. 653, 65–79 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  • Cunha, A.W., de Lima, E.L., de Lima, H.F., Lima Jr., E.A., Medeiros, A.A.: Bernstein type properties of two-sided hypersurfaces immersed in a Killing warped product. Studia Math. 233, 183–196 (2016)

    MathSciNet  MATH  Google Scholar 

  • Dajczer, M., Hinojosa, P., de Lira, J.H.: Killing graphs with prescribed mean curvature. Calc. Var. Partial Diff. Eq. 33, 231–248 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Dajczer, M., de Lira, J.H.: Entire bounded constant mean curvature Killing graphs. J. Math. Pures Appl. 103, 219–227 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  • Dajczer, M., de Lira, J.H.: Entire unbounded constant mean curvature Killing graphs, to appear in Bull. Braz. Math. Soc. (2016). doi:10.1007/s00574-016-0019-3

    Google Scholar 

  • Emery, M.: Stochastic Calculus on Manifolds. Springer, Berlin (1989)

    Book  MATH  Google Scholar 

  • Grigor’yan, A.A.: Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds. Bull. Am. Math. Soc. (N.S.) 36, 135–249 (1999)

  • Grigor’yan, A., Saloff-Coste, L.: Dirichlet heat-kernel in the exterior of a compact set. Comm. Pure Appl. Math. 55, 93–133 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  • Gromov, M.: Isoperimetry of waists and concentration of maps. Geom. Funct. Anal. 13, 178–215 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  • Impera, D., Rimoldi, M.: Stability properties and topology at infinity of \(f\)-minimal hypersurfaces. Geom. Ded. 178, 21–47 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  • Lichnerowicz, A.: Variétés Riemanniennes à tenseur C non négatif. C. R. Acad. Sci. Paris Sér. A-B 271, A650–A653 (1970)

    MATH  Google Scholar 

  • Lichnerowicz, A.: Variétés Kählériennes à première classe de Chern non negative et variétés Riemanniennes à courbure de Ricci généralisée non negative. J. Diff. Geom. 6, 47–94 (1971)

    Article  MATH  Google Scholar 

  • Morgan, F.: Manifolds with density. Notices Am. Math. Soc. 52, 853–858 (2005)

    MathSciNet  MATH  Google Scholar 

  • O’Neill, B.: Semi-Riemannian Geometry with Applications to Relativity. Academic, London (1983)

    MATH  Google Scholar 

  • Pigola, S., Rigoli, M., Setti, A.G.: A remark on the maximum principle and stochastic completeness. Proc. Am. Math. Soc. 131, 1283–1288 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  • Pigola, S., Rigoli, M., Setti, A.G.: Maximum principles on Riemannian manifolds and applications. Mem. Am. Math. Soc. 174, 822 (2005)

  • Rimoldi, M.: Rigidity results for Lichnerowicz Bakry-Émery Ricci tensors. Ph.D. thesis, Università degli Studi di Milano, Milano (2011)

  • Stroock, D.: An Introduction to the analysis of paths on a Riemannian manifold. Math. Surv. Monogr. Am. Math. Soc. 4, 59 (2000)

    Google Scholar 

  • Udrişte, C.: Convex Functions and Optimization Methods on Riemannian Manifolds. Springer, The Netherlands (1994)

    Book  MATH  Google Scholar 

  • Vieira, M.: Harmonic forms on manifolds with non-negative Bakry-Émery-Ricci curvature. Arch. Math. 101, 581–590 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  • Wu, J.Y.: \(L^p\) Liouville theorems on complete smooth metric measure spaces. Bull. Sci. Math. 138, 510–539 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  • Wei, G., Willie, W.: Comparison geometry for the Bakry-Émery Ricci tensor. J. Diff. Geom. 83, 377–405 (2009)

    MATH  Google Scholar 

  • Yau, S.T.: Harmonic functions on complete Riemannian manifolds. Comm. Pure Appl. Math. 28, 201–228 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  • Yau, S.T.: Some function-theoretic properties of complete Riemannian manifolds and their applications to geometry. Indiana Univ. Math. J. 25, 659–670 (1976)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The first author is partially supported by CNPq, Brazil, grant 303977/2015-9.

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Correspondence to Márcio S. Santos.

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Lima, H.F.d., Lima, E., Medeiros, A. et al. Liouville Type Results for Two-Sided Hypersurfaces in Weighted Killing Warped Products. Bull Braz Math Soc, New Series 49, 43–55 (2018). https://doi.org/10.1007/s00574-017-0043-y

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  • DOI: https://doi.org/10.1007/s00574-017-0043-y

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